Lesson 16

Interpreting Inequalities

16.1: Solve Some Inequalities! (5 minutes)

Warm-up

This warm-up is an opportunity for students to recall understandings and techniques from the previous lesson.

Launch

Optionally, provide access to blank number lines.

Student Facing

For each inequality, find the value or values of \(x\) that make it true.

  1. \(8x+21 \leq 56\)
     
  2. \(56 < 7(7-x)\)
     

Student Response

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Anticipated Misconceptions

If students express the solution in words or by graphing on a number line, applaud their use of these representations. Encourage them to attempt to express the solution using the efficient notation, as well. Direct their attention to any anchor charts or notes that remind them of the meaning of the symbols involved.

Activity Synthesis

Ask one student to share their process for reasoning about a solution to each problem. Address and resolve any discrepancies that arise.

16.2: Club Activities Matching (10 minutes)

Activity

In this activity, students analyze four situations and select the inequality that best represents the situation. (In the activity that follows, students will work in small groups to create a visual display showing the solution for one of these situations.)

Launch

Tell students that their job in this activity is to read four situations carefully and decide which inequality best represents the situation. In the next activity, they will be responsible for writing a solution for one of these situations. Give 5–10 minutes of quiet work time.

Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, use the same color to highlight key words or phrases in the situation with its corresponding inequality sign in the matching inequality.
Supports accessibility for: Visual-spatial processing
Reading, Representing: MLR2 Collect and Display. As students work, circulate and collect examples of words and phrases students use in their written response to "Explain your reasoning" for each question. Look for different ways students describe what the variable represents, how they know which number is the constant term, how they know which number should be multiplied by the variable, and the direction of the inequality symbol that makes sense for each context. Organize the phrases for each of these considerations and display for all to see. This will help students to focus on all of the important elements of the inequality they are assigned in the next activity, with language they can use in small group discussions.
Design Principle(s): Support sense-making; Maximize meta-awareness

Student Facing

Choose the inequality that best matches each given situation. Explain your reasoning.

  1. The Garden Club is planting fruit trees in their school’s garden. There is one large tree that needs 5 pounds of fertilizer. The rest are newly planted trees that need \(\frac12\) pound fertilizer each.
    1. \(25x + 5 \leq \frac12\)
    2. \(\frac12 x + 5 \leq 25\)
    3. \(\frac12 x + 25 \leq 5\)
    4. \(5x + \frac12 \leq 25\)
  2. The Chemistry Club is experimenting with different mixtures of water with a certain chemical (sodium polyacrylate) to make fake snow.
    To make each mixture, the students start with some amount of water, and then add \(\frac17\) of that amount of the chemical, and then 9 more grams of the chemical. The chemical is expensive, so there can’t be more than a certain number of grams of the chemical in any one mixture.
    1. \(\frac17 x + 9 \leq 26.25\)
    2. \(9x + \frac17 \leq 26.25\)
    3. \(26.25x + 9 \leq \frac17\)
    4. \(\frac17 x + 26.25 \leq 9\)
       
  3. The Hiking Club is on a hike down a cliff. They begin at an elevation of 12 feet and descend at the rate of 3 feet per minute.
    1. \(37x - 3 \geq 12\)
    2. \(3x -37 \geq 12\)
    3. \(12 - 3x \geq \text-37\)
    4. \(12x - 37 \geq \text-3\)
       
  4. The Science Club is researching boiling points. They learn that at high altitudes, water boils at lower temperatures. At sea level, water boils at \(212^\circ \text{F}\). With each increase of 500 feet in elevation, the boiling point of water is lowered by about \(1^\circ \text{F}\).
    1. \(212 - \frac{1}{500}e < 195\)
    2. \(\frac{1}{500}e - 195 < 212\)
    3. \(195 - 212e < \frac{1}{500}\)
    4. \(212 - 195e < \frac{1}{500}\)

Student Response

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Activity Synthesis

At this time, consider not validating which inequalities are correct. When students get into groups for the next activity, they can compare their responses with the members of their groups and resolve any discrepancies at that time.

16.3: Club Activities Display (20 minutes)

Activity

In this activity, students interpret parts of an inequality in context, term by term; for example, what quantity must \(\frac{1}{2}x\) represent? Then they make sense of the entire inequality by thinking about what question would be answered by the solution to the inequality. Notice groups that create displays that communicate their mathematical thinking clearly, contain an error that would be instructive to discuss, or organize the information in a way that is useful for all to see. At this point, there is very little scaffolding for the solving of the inequality itself. 

Launch

Arrange students in groups of 2–3 and provide tools for making a visual display. Assign one situation to each group. Note that the level of difficulty increases for the situations, so this is an opportunity to differentiate by assigning more or less challenging situations to different groups.

Engagement: Develop Effort and Persistence. Provide prompts, reminders, guides, rubrics, or checklists that focus on increasing the length of on-task orientation in the face of distractions. For example, create an exemplar display including all required components, highlighting different ways to communicate mathematical thinking clearly.
Supports accessibility for: Attention; Social-emotional skills

Student Facing

Your teacher will assign your group one of the situations from the last task. Create a visual display about your situation. In your display:

  • Explain what the variable and each part of the inequality represent
  • Write a question that can be answered by the solution to the inequality
  • Show how you solved the inequality
  • Explain what the solution means in terms of the situation

Student Response

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Student Facing

Are you ready for more?

\(\{3,4,5,6\}\) is a set of four consecutive integers whose sum is 18.

  1. How many sets of three consecutive integers are there whose sum is between 51 and 60? Can you be sure you’ve found them all? Explain or show your reasoning.

     
  2. How many sets of four consecutive integers are there whose sum is between 59 and 82? Can you be sure you’ve found them all? Explain or show your reasoning.
     

Student Response

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Activity Synthesis

Select groups to share their visual displays. Encourage students to ask questions about the mathematical thinking or design approach that went into creating the display. Here are questions for discussion, if not already mentioned by students:

  • How did you figure out what the \(\frac{x}{7}\) term represents?
  • How did you decide on the direction of the inequality for the solutions?
  • Did anyone with the same problem do one of the steps differently? Share what you did differently so we can learn from what happened.
  • How do you know there are 25 pounds of fertilizer available?

Alternatively, have students do a “gallery walk” in which they leave written feedback on sticky notes for the other groups. Here is guidance for the kind of feedback students should aim to give each other:

  • What is one thing that group did that would have made your project better if you had done it?
  • What is one thing your group did that would have improved their project if they did it too?
  • How did the group decide the direction of inequality for the solutions?
  • Does their answer make sense in the situation?
  • Is their mathematics clear and correct?
  • If there was a mistake, what could they be more careful about in similar problems?
Representing, Conversing: MLR7 Compare and Connect. During the launch, make sure at least two groups are assigned to each situation (assign fewer contexts if there are fewer than 8 groups). Assign groups who worked on the same situation to review each other's display. Ask groups to look closely at how the inequality was solved, then to identify and discuss what is the same and what is different, compared to their own display. If the other group’s solution is the same, students should compare the strategies used. If the solution is different, students should look any errors in reasoning, either in their own or the other group’s method. Ask each group leave a comment on a sticky note that describes the comparison they discussed. This will help students make sense of the reasoning of others by interpreting work that is similar to their own.
Design Principle(s): Maximize meta-awareness; Support sense-making

Lesson Synthesis

Lesson Synthesis

In this lesson, we saw how inequalities can be applied to real-world situations. Some questions to bring this work together:

  • Suppose your friend asks you to write some practice problems for solving inequalities. You want to write an inequality that has a solution of \(x\leq\text-8\frac23\). Describe how to write such an inequality. 
  • Think about an after-school activity in which you are involved. Write an inequality that represents a situation related to that activity. Be prepared to share the inequality and an explanation of its terms with the class.

If time allows, have students solve their inequalities.

16.4: Cool-down - Party Decorations (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

We can represent and solve many real-world problems with inequalities. Writing the inequalities is very similar to writing equations to represent a situation. The expressions that make up the inequalities are the same as the ones we have seen in earlier lessons for equations. For inequalities, we also have to think about how expressions compare to each other, which one is bigger, and which one is smaller. Can they also be equal?

For example, a school fundraiser has a minimum target of \$500. Faculty have donated \$100 and there are 12 student clubs that are participating with different activities. How much money should each club raise to meet the fundraising goal? If \(n\) is the amount of money that each club raises, then the solution to \(100+12n=500\) is the minimum amount each club has to raise to meet the goal. It is more realistic, though, to use the inequality \(100+12n\geq500\) since the more money we raise, the more successful the fundraiser will be. There are many solutions because there are many different amounts of money the clubs could raise that would get us above our minimum goal of \$500.